a,
ty
a,
NN
ity
ry
Ra

TECHNICAL REPORT

“POWER SPECTRUM ANALYSIS
OF
INTERNAL WAVES FROM
OPERATION STANDSTILL

Applied Oceanography Branch

Division of Oceanography

OCTOBER 1955

i
TH?
fo. TR Xt U. S. NAVY HYDROGRAPHIC OFFICE
WASHINGTON, D. C.

TR-26

ABSTRACT

Temperature fluctuations in the ocean obtained from data
collected during Operation STANDSTILL are analyzed by
power spectrum techniques to determine the periods of
internal waves. The analysis indicates that the power
outside the noise level is concentrated in periods from 20
to 26 hours. Two physical causes for diurnal periods are
suggested: eddies due to instability south of the Gulf
Stream, and free oscillations, controlled by the Coriolis
force, in resonance with the diurnal tidal components. Itis
concluded that the latter cause probably is responsible for
the peak power of internal waves at periods between 22
and 24 hours.

written by
ALLEN L. BROWN, EDWARD L. CORTON
and
LLOYD S. SIMPSON

Applied Oceanography Branch
Division of Oceanography

FOREWCRD

Many phases of naval operations involve the application of
oceanographic principles. In order to predict changes in oceano-
graphic parameters, it is necessary to understand the nature of
the physical processes that are producing the changes. Accurate
methods of predicting changes which occur in the thermal structure
of the ocean cannot be obtained without consideration of internal

Waves e«

This report presents the results of research concerning the
existence and periods of internal waves in an area northeast of
Bermudas Activities receiving this publication are requested to
forward their comments to the Hydrographic Office.

| i oe

J. Be COCHRAN
Captain, U. S. Navy
Hydr ographer

Hmm mnen AON

alata

DISTRIBUTION LIST (D)

CNO (Op=31, 314, 316, 533)
BUAER (2)

BUCRD (2)

BUSHIPS (2)

BUDOCKS (2)

ONR (Code 16, 63, 65, 66)
NOL (2)

NEL (2)

NRL (2)

DIMB (2)

COMINLANT (2)

COMENPAC (2)

COMHUKLANT (2)

COMOPDEVFOR (2)
NAVMINCOMEASTA (2)
NAVMENWARSCOL, Yorktown (2)
MHU (2)

NAVWARCOL (2)
USNAVSUBSCOL (2)

USNUSL (2)
NAVPOSTGRADSCOL, Monterey (2)
GIA (2)

BEB (2)

ARGWA (2)

ASTIA (2)

WHOL (2)

SIO (2)

pire)

TEXAS A&M (2)

UNIV WASH (2)

NYU (2)

MIT (2)

iv

Foreword OO) 8) 0) ke e @ e o @

CONTENTS

Dbl roby walvosg) Melissa) MG NG G0 oda sitctifejiiejira 6606
PUIBURES ANG <a Vere els ei) oi ve Meh toluene iauiiofilie <0) “© ah (ilo
I. botwatelewbiomablesa GY OH 5) 6) Ovo Gp OOO G20 6
II. Ita SneoeUL WEES GC “di 6 6 685 Oh OOOO 6 a0
He me AMA STS ON WaGay else) lel verlleiifo/ Vell (rin le)is) (rel! of i's
IV. Resume SmMotiNaailey Susie elite Mol Melts rel lalate eae
We Eddy Motion in the Area of Operation STANDSTILL

A, Eddies Resulting from Instability . . ..

B. Inertia Waves 6

VI.

lspllollsiowaeaideiy Gag 5 loG 0 Ob o.oo, oo OO OOO 6

°

a

®

°

summary and Conclusions: oo, ses pire oe «

aialal

|}
s

FIGURES

Frequency Distribution of Depth of 66° F. Isotherm. ©
Autocorrelation Function for Hourly Readings . . « o «
Power Density Functions for Hourly Readings . . . » e
Power Density Functions for Half-Hourly Readings . . o
Velocity Section Across the Florida Current. . ..-. ce

Velocity Vectors and 200-Meter Isotherms in the Gulf
Stream off Woods Hole, Masse, May-June 1953 ...-+o

Graphical Representation of Harizontal and Vertical

° we Oo xs x
Motion and Displacement of the 66° i. Isotherm for
HENS dacoorserains Westie) 5 5 5 05 0 0 6 6 6 0900 0

TABLE

Frequency of ‘Occurrence of 66° F, Isotherm in Class
Intervals of Depth Oey Ono SCO) TOM Or Or DO On OHO O0e uw 9G oO

Vi

Page

2

I. INTRODUCTION

In June and July 1953 Operation STANDSTILL was conducted in the area
northeast of Bermuda. During the course of this operation an anchor station
was maintained at 33°33'N, 62°25'W for a period of 25 days, from 1] June to
6 July. Bathythermograph readings were taken every half hour as part of the
oceanographic observations made at the anchor station. The following dis-
cussion deals with one aspect of these observations: the analysis of the
wave spectrum of internal waves.

II. INTERNAL WAVES

Previous discussion of the theory and existence of internal waves in
the ocean has been directed toward the demonstration of the occurrence of
internal waves of tidal period. Thus, Haurwitz (1953) has examined the
temperature data from Meteor stations 385 and 38. and the Altair anchor
stations and has concluded that the oscillations at Meteor station 385
show possible existence of internal waves of semidiurnal tidal period.

_ With the recent development of the techniques of power spectrum
analysis (Pierson, 1952), it is possible to analyze the spectral components
of any suitable series of data comprising a Gaussian process. If it is
assumed that the fluctuations of temperature in the ocean represent a
stationary Gaussian process, the power spectrum analysis will indicate
the significant periods present in the series and the power associated with
these periods. In this way, it is possible to determine whether internal
waves of tidal periods exist, as well as other pericds greater and smaller.

III. ANALYSIS OF DATA

An examination of all the BI data showed that the deepest continuous
isotherm was 66° F. Accordingly, the depth of this isotherm was determined
from each half=hourly bathythermogram. These data are presented in table
1. Two ships in succession participated in this operation, the USS RE-
HOBOTH and USS SAN PABLO. The frequency of the readings of the depth of
the 66° F. isotherm is given for 25-foot intervals from 76 to 1,100 feet.
This frequency distribution is graphed in figure le

Theoretically, distributions which are to be analyzed by power spectrum
methods should be normal, i.e., Gaussian (Tukey, 199). The present dis-
tribution has an upper limit at the surface and therefore is of the Pearson
Type III. A normal distribution curve was fitted to the observed distri-
bution, taking only the depths from 76 feet to 575 feet. The resulting
curve is shown in figure 1 as a dashed line. The fitted curve indicates
that the observed curve does not depart greatly from the normal distribu~
tion. Thus, since the frequency distribution is nearly normal, power
spectrum analysis by the usual methods will not give meaningless results.
The series of 1191 readings was analyzed for its power spectrum (Fierson
and Marks, 1952) in five different ways:

1

TABLE 1. FREQUENCY OF OCCURRENCE OF 66° F. ISOTHERM
IN CLASS INTERVALS OF DEPTH

FREQUENCY
Fitted

Depth USS USS Total Frequency
fr REHOBOTH SAN PABLO Observed (Unadjusted )
76~100 0) 2 2 13
100-125 ) 3 3 25
126-150 5 2 7 38
151-175 25 2 27 57
176-200 52 36 88 80
201-225 Xe) cal 101 98
226-250 70 75 16 119
251-275 26 78 10) 129
276300 67 9) 161 128
301-325 hs 54 101 118
326-350 59 5 10) 99
351-375 5) 22 77 hit
376-100 52 35 83 Sh
01-25 Wy 5 19 37
h26-50 16 12 23 22
51-75 9 4 13 als)
76-500 Uy, 23 37 6
501-525 aL 3 Ly 3
526=550 2 1 3 1
551-575 ) 6) P)
576-600 0 1h 1
751-775 (0) 2 2
776-800 0 5 6
801-825 © (9) 9)
826-850 0) dl a
851-875 0) 2 2
876 ) h h
901-925 0 ) h
926-950 0 h in
951-975 fo) 3 3)
976-1000 0) 0 9
1001-1025 ©) h h
1026-1050 0 L h
1051-1075 0 a a
1076-1100 0 ) h
Total 563 628 1191. 1116

Note: Unadjusted total fitted frequency of 1116 compares with observed
total of 1113 in range from 101 to 550 feet.

2

OL

os

09

oye

08

06

Oot

Ort -

ozt -

O€T

oot
OLI

O8T

o6T

WHSHLOS! “3099 3O Hldad JO NOILNEIYLSid AONANOAYS ‘| 3yuNdIS
0

(4984) Hidad
‘ -
f
/
/
i
/
\
\
\
/

OOcI OSII OOII OSOIT OOOT ose 006 oss oos os ooL 0s9 0090S Sie OO Seen: eo
OOZI OSII OOII OSOI OOO os6 006 O0S8 008 osZ ooL 0S9 009 OSS 00S OSt OOv OSE oO0€ OS O02 OSI foley e os i)

oot os i0)

\

ee

og

ee a aes Be | STI
h
\

06

oot

Ort

oct

OET

Ovt

Ost

NOILNSIYLSIG TIWANYON Q31114a WOU
AONSNOSYS WOILSHYOSHL — —
AINANOSYS WLOL GSAYXSSEO

QN3937

ogt

OZt

ost

o6t

SNOILVAYSSSO 4O YSEWNN

1) Hour®y readings were used for computation, starting with the first
observation.

2) The remaining hourly readings were used for computation, starting
with the second observation, in order to estimate the variability of hourly
samplings.

3) and ) The first and second halves cf the BT data, representing the
observations made by the USS REHOBOTH and USS SAN PABLO respectively, were
analyzed using the half-hourly readings.

5) The whole of the half-hourly BT data was analyzed. In each case
60 lagsiwere usede

One preliminary result of the power spectrum computation is the auto-
correlation function. The autocorrelation functions for all five of the
series were practically identical and are not shown separately here. The
function presented in figure 2 is for the hourly readings commencing with
the first observation. It is evident that the major cyclical components are
found in the band of periods centered at 2); hours.

The results of the power spectrum analyses are graphed in figure 3
for the hourly readings and in figure }; for the half-hourly readings. Since
a total of 60 lags was used in all analyses, the number’ of degrees of freedom
available was about 19 for the first four analyses and 39 for the last analysis.
Thus, the accuracy of the analyses is somewhat lower than it would be with a
longer series of observations. Confidence limits are not shown in the figures.

IV. RESULTS OF ANALYSIS

The spectral analysis derived from the 66° F. isotherm shows that most
power is concentrated in the band from 20 to 30 hours. This result was fore=
shadowed by the autocorrelation function in figure 2 which is markedly cyclical
with peaks at multiples of 22 to 26 hours. The semidiurnal period to be ex=
pected from tidal motions is apparently neglibible. Although in figure 3 the
small peak between 10 and 12 hours appears to coincide with the tidal period,
the autocorrelation function shows that this peak is probably due to the short
length of record. The noise level of figure 3, therefore, must be of the order
of 20,000 fte@sec. In figure l, there is no outstanding power peak in the free
quency range in which definition is good. The great differences between the
spectra from the two ships, taken at different times, indicate that the noise
level for the half-hourly readings is also about 20,000 fte“sec.

It is thus likely that the whole spectrwn below 20 hours represents
white noise with no important spectral components. For an observational in=
terval of one-half hour, the spectrum is best estimated in the range from 1
to 10 hours; for observational periods of one hour, adequate definition in
the spectrum exists in the range from 2 to 30 hours.

4

SONIGVSY ATYNOH HOS NOILONNS NOILVISYHYOIOLAV ‘2 3YnNdoIS
csanoy) O94

09 8S 9S S 2G OS 8% 9 vy 2y Ov BE OE vE ZE OF Ce" OS U4 44 Oe Gh Oi wi ei OK ESS oe v Cle O

LN3ID1S4S3309 NOILVISYYNOS

‘SONIQVSY ATYNOH NOS SNOLLONAL ALISNEG
(s4nou) GOluad

S9V1 30 ON
OSREESSIE OSD Gmc Cm OG SY OTL Vaict OLE mSEmeSS = Vol Che Of 8c, 9c ies cca Oc
[ : i

S: 9

Eo es oS

(1LW9 SUNOH N3AS) Ir SONIGV3Y ATuNOH-—- — —

(YNGH 1SVd “NIN Of} 7 SSONIGVSYS ATENGH: =
QN3937
ot eo tee? Ee eee eal ae alte ase elle t |) SMe S|

HAMOd € |
8 OL

8t oT mt
aK

Saundia

ALISNAG YAMOd

(998,43 30 S000) (ny)

SONIGVSY ATYNOH—-S1VH YOS SNOILONNS ALISNSGQ YSMOd ‘pv AYNdIS
(sanoy) doly3ad

O'l : St € v a) OI Sst o€ ©
S9V1 JOON
09 OS' Sb 97 YY ch Or 38E2 98 ve “CE OF 82 9c ¥e ce 0c BI
ool
O18Vd NVS SSM SONIGWSY ATYNOH—41VH ane
HLOSOHSY SSN SONIGVAY AIYNOH-JS1VH ——
Oz
NOILVY¥SdO SYILNS Y¥OS SONIGVSY ATYNOH-431VH ——
O€T
QN3931
ort

nv) ALISNAGQ YAMOd

(

000)

"90S - o Ss
( saad

In order to obtain a very accurate power spectrum analysis of waves
of periods from 10 to 30 hours, which include the tidal periods, the time
interval would have to be 3 to 5 hours and the length of the period, therefore,
at least 3 to 5 times as long as the present records, i.eey|from 75 to 125 dayse
Such a record could be obtained at a fixed weather ship location if regular
bathythermograph readings were taken there.

The presence of aliasing in the record may be important. Pierson (1952)
defines aliasing by saying that if the observational interval is too large,
the power per band width can have other values of power from other frequencies
aliased into (or added into) the true values for the particular band desired.
If there is any large amount of power associated with periods of less than
one hour in the present case, the spectral analysis is in error by that amotnt.
Schule (1952) has discussed the existence of internal waves with periods of
minutes. In order to show that internal waves with such periods have ap~
preciable power, it would be necessary to have records taken every minute,
comprising a large number of observations with lags of 200 to 500 taken into
account. However, if the power associated with internal waves of periods
less than one hour is low,\ie@€ey in the noise level, it then becomes possible
to consider that the spectrum is white below about 20 hours and that the
conclusions of the preceding paragraphs. apply.

The mean depth of the 66° F, isotherm was 32) feet and its standard
deviation 87 feet. The mechanism which caused approximately diurnal periods
in the isothermal depth could not have been vertical heating, for two reasons:
the depth is too great, and the isotherm was located below the thermocline,
which was fairly sharpe A horizontal pulsation of water of different tempera-~
tures could have caused.the observed change. The average diurnal change in
the depth of the isotherm was about 150 feet. It remains therefore to find
the mechanism which caused diurnal advection of heat into the area of Operation
STANDSTILL.

V. EDDY MOTION IN THE AREA OF OFERATION STANDSTILL

A. Eddies Resulting From Instability

As a result of a survey of the Gulf Stream made almost simultaneously
with Operation STANDSTILL, it has been suggested that regular pulses in the
‘Gulf Stream may be initiated by the diurnal tide of the Gulf of Mexico (Von
Arx, Bumpus, and Richardson, 195). At intervals of 2) hours the tidal motion
may set into action jets of fast-moving water and produce noticeable fluctuae=
tions in the structure of the Gulf Stream. On the south side of the stream,
it may be postulated that eddies, set up in phase with the inertial period,
influenced the area of Operation STANDSTILL. Each eddy contained a jet of
warmer water in swift motion followed by a slower moving current of cooler
water. As this alternately warm and cool water passed through the area of
the operation, the isotherms moved upward and downward in response to the
water movement.

The regular pulses in the Gulf Stream may be compared mathematically to
perturbations in a geostrophic flow. If a geostrophic flow is perturbed,
the flow vector will show an acceleration which will cause a particle to
feviate from geostrophic flow. This deviation may be represented as

8

(1)
where
f m the Coriolis parameter,
V = the velocity vector,
Vg = the geostrophic velocity vector, and
kL s the unit vector in an upward direction.
Subs ti tu ting

VeiVee merece
Va = Vqo* d Vz e Vg. + (dE -V) Vg.

since every velocity vector may be represented by the initial velocity
plus an increment due to the acceleration, the resulting equation is

—ez av ses oy a
dt FLV +3 St-Vy. (43°F) Vg] *k s (2)
Separating this equation into the components of velocity u and v gives

d : > dv
Se =f Vo-Vge~ (44 Max - PE dy) TH

ie
ahs
v)

FEC) ay cee lu eae te

where
dx = FPerturbed displacement eastward, and

«y= Perturbed displacement northward.

—P —»P

In equations 3 and ) the unit vectors t and Jj are directed toward
the east and north respectively, so that the equations are the Pantone
of motion in the two directions,

The increment of energy associated with the displacement may be repre-
sented as

dv :
ye = Se aos are dup (5)

Substituting the values of gu and ay from equations 3 and | gives

£ (SPF) lagh-(S8-#) de) +B PB) aya].

This devel opment is a special application to incompressible LInids of the
more general exposition by Van Meighem (1951). Equation 6 contains the
criterion for stability. Assuming flow from west to east (so that the
only perturbations will be in the y-direction), the last two terms of the
equation will drop out, and the stability depends on the sign of the first
factor.

ie 2 Ug. 3) (Way the perturbed energy of the
Oy nO or sa a

\

Bp i ou
system is increasing and the system is unstable; if “OR, Cai aoe the
system is stable. Therefore, the equilibrium ccndition is attained when

2 Uge ie
f s Lusing « Since the latitude of Oneration STANMSTIT, ves 33°33!N,

sin @ is 0.553, and the eanilibrium or inertial period is therefore 2h
hours divided by 2 times 0.553, or 21.7 hours.

Equation 6 may be used to show that the maximum possible velocity gradi-
ent toward the right-hand side of a west~to-east flow must be smaller than the
Coriolis Parameter (f) or else the flow becomes unstable and eddies form.

On the left-hand side of the current, however, an increase in shear merely
adds to the stability. Two illustrations show how the Gulf Stream system
conforms to these principles. figure 5, taken from Worthington (1955), is
_a velocity section across the Gulf Stream off Woods Hole, Mass. This figure
clearly indicates the spreading out of the current toward the right, which
is necessary to prevent instability, whereas the left side of the current
shows the packing of the jet.

SETTLEMENT PT.
PALM BEACH AAMAS fs
5162

6 |
200]
300 |
400|
500;

600 |

io 20 |
NAUT. MILES \

700 |

FIGURE 5. VELOCITY SECTION (cm/sec.) ACROSS THE FLORIDA
CURRENT, 7 FEBRUARY 1954. THE SMALL NUMBERS ON
THE LEFT-HAND SIDE OF THE CURRENT ARE SALINITY

ANOMALIES. (EXPLANATION IN TEXT).

ah

Figure 6 is taken from Von Arx, Bumpus, and Richardson (195). In
this figure the velocities of the Gulf Stream measured just three weeks
previous to Operation STANDSTILL are shown by arrows. Three eddies are
delineated by the clockwise turning of the vectors at distances of 30
to 50 miles away from the jet and on the right-hand side of the flow.

It is evident from the arrows on 29 May that previous to the formation

of an eddy on 31 May the vectors had a northward component, and the flow
must then have been stable. The data do not allow determination of the
exact period of formation of the eddies because of the sporadic nature

of the observations. The above development leads to approximately the
right period but has a serious flaw in that the radius of the eddy must
be small because the entire mass of water is in motion. Inasmuch as the
observed currentsin the vicinity of the anchor station averaged less than
0.5 knots and were not regular in direction, any existing eddies must have
had a radius of less than 2 miles. However, the radius on which the ship
swung at anchor (disregarding drag on the bottom) was a minimum of 2.5
miles, because 20,000 feet of anchor cable were payed out in a depth of
900 fathoms. It is therefore evident that the phenomenon which caused the
regular change in isothermal cGepth shown by the spectral analysis regardless
of the location of the ship on its anchor arc was much larger in diameter
than the small instability eddies.

B. Inertia Waves

; A second form of motion that may be considered is inertia waves con-

trolled by the Coriolis force. The equations of motion for these waves
considering a finite depth h and postulating that the vertical velocity w

is equal to zero both at depth h and at the surface (z = 0), may be written as

Ou Seco eal a

St * W+ 2a cosh - fu tox =O,

ov a

at tfu =o

aw rs) = p7) \

St LWOuUcsd +H 35 =O (7)
where ole 5 is the svecific volume. These €quations are similar to

those on p. 209 of Haurwitz (19l)1) except that in the present case, the inertial
motion is given for all latitudes instead of only at the poles. Equations (7)
do not contain a gravity term because the water is considered to be homo-~
geneous and the motiyn thus entirely inertial.

The equation of certinuity completes the initial set of conditions:

12

ES6! ANN -AV “SSYW ‘3I0H SGOOM 440
WVSHLS SIND SHL NI SNYSHLOS! YSLIN-002 GNV SYOLDGA ALIOONSA 9 3yNNOdI4

3nnr

pa 02

1
em

OS Ol
2 oz
“do SUNLVU3dW3L ¥313W 00z oe
o¢
B Z
| Ov
z
4 | rai ats
| : \ Ve He)
i | f Ne re
, “ g Wo SNEAY {
j | i ail 5
LS a8 me ee
~ SY a | lec ian | oz
A aie a xl j | leas lied ‘ eae
EES ett ewes eS
: i Car} — i | j
PAN tee |) al ES fe
TAN WS mS H | Po ie Bi ee
ane ean pe a. a O
Se yy ae hee et Ee Poe |
alah Sy yeas } ST -- eel es eae
TS ee Hl ee ws | 2 N._|
Pod We = ea | ; 1]
j i al D3S/WI 301X SbEezI-0 4 ia ace i \
[5S I al 2. QUALI : s} * i “ poz
; { 4 (39V4UNS) ALIDOTSA ~~ 3-9 | i!
nN ; | .
° { Oe
|
i pov
K

au, WwW _o

a4 az (8)

for east-west motion.

Considering a free oscillation of inertial type, the wave particle
velocities in the x, y, and % directions are:

BI!) ON oe nm DR ok ame )

= ei lkx+Ib)(a aos ie D sin nme |

h (10)
be iCkex+5t) _. nTlze

w= Ee sin (12)
where QO = = is the frequencye
To these must be added the driving force

‘ +6 6 z
E = Be (E2s eS Ee le
h h (12)

The solution of equations 9, 10, 11, and 12 for the unknowns A through
G is accomplished by substitution in (7)'and by the combination of terns
to se eparate the coefficients of the sine and cosine factors. Since the

Sine and cosine are linearly independent, their sum must be zero. Hence,

(ic A- =I + ike eos Gt = 4Qwcosd: Sap gf [b— -£D+iak@)sinn— h FO us)

(fAtire)cs™ +(fe+4iadD) sin TF =O Qh)
ANT

Ce aGe Zui cos: A) cos Go -(2FF E+ Iwo ¢ e-is€) sin oe ae On and (2
hig a) cos aie LKB sin AF 26 05)

14

Theretore, the following equat* uns must be satisfied:

fa-LSA-LXKF =0

(17)
fD-LIB-2wWcs dE - Lake =o (18)
fA+LOG=o (19)
f8 +ticgD=0 (20)
ZWasd- A-x PT G@=0 (21)
Qn”
om Thoin ~ LOE 1 ae =O (22)
LkA+ Beso. and £03)
LKB=O, (2h)
From (2),) and (20), it is evident that
B —— i De
Solving (19), (21), (22), and (23) in terms of A gives
aE
E=-l ng A (25)
as cza (26)
G = 222 aos4-A 4 ana (27)
F == {= £5 A ° (28)

Substituting (23) and (21) in (18) gives the identity

Za) cos dq. ey ~ tk gana

proving the consistency of the solution. Next, suvstituting (23) and (17)
into (22) gives

nt i wie ‘ (29)

ok? h® = nem(f7-c7) ,

[i+ (224) ]? (30)

since b= ar

In shallow water, i.e, when bh << ts 9 s=f » and it is seen
that the period would be approximately the same as in the case of instability
eddies, or 21.7 hours. However, since h/L is positive, even though small,
the denominator in (30) is greater than 1, and hence rans is less than
£. The period is then somewhat greater rev 21 ef hours. !

The physical effect of the inertial motion described above may be seen
by considering the motion in the horizontal (xy) plane. In this case, for
a fixed z, equations (9) through (12) reduce to

vs eos (kx +t) * A cos a
V = sin Ckx+0t) * @ cos ote (21)

p = —sin Cke+Ot)> H(z) ,

16

The pressure gradient in the x direction is

SE = —k H(z) cos Ckx+t) . (32)

Hence,
a yo
oe a ae =| e
27.2 nile 2 o ne
A> cos ne G?cos* TT (33)

Thus, the hodograph with respect to time is in the form of an ellipse, and
also the particle motion is elliptical in the xy plane. The horizontal
particle path is shown in figure 7. The horizontal and vertical motions

are represented schematically in figure 7. The figure consists of three
parts. The uppermost represents the ellipses of motion near the surface

and near the bottom, which are 180° out of phase. The middle portion of

the figure shows an east-west vertical section with one wavelength repre-
sented. At the points marked with plus and minus signs the flow is out of
and into the paper respectively. Three pressure surfaces, P,, Po, and P3,
indicate the perturbation pressure gradients. It will be noted Sper Po is
shown somewhat nearer the bottom than the corresponding depth h/2, and that
for this pressure there is no horizontal pressure gradient and thus no hori-
zontal motion; all the motion is vertical at this point. This follows since
the second term in equation (12) is not zero at h/2, so that the minimum per~
turbation pressure P = 0 is found at some z = h/2 plus a small amount. The
third part of the figure shows the displacement of the 66° F. isotherm is
relation to the wave propagation. In actuality, the wavelength is essen=
tially that of the earth's circumference, so that only a portion of one cell
is found in the Atlantic Ocean. However, the complete cycle passes each

_ point in the ocean during a lunar day as the wave propagates westward.

Figure 3 may be reexamined in the light of the above findings. It is
suggested that since the inertial period is 21.7 hours plus an unknown amount
depending on the ratio of the depth to the wave length (from equation 30),
the spectral bands center:d at 20 and 2h hours show the preatest power.

There is a sharper peak in the band from 21.8 to 26.7 hours possibly be-
cause the diurnal tide acts in resonance with the inertial force. Corres-
pondingly, it is possible that no semidiurnal period shows on the power
spectrum because the inertial filter is "transparent" to waves of 12-hour
periods iee., the natural period (21.7 hrs.) of the inertial oscillations
iS unresponsive to the 12-hour tidal constituent.

in the case of stratified water, where a component due to gravity
modifies g* by analysis similar to that given above, it can be shown
R

that, GO for the free wave is given by the following approximate relation:

17

GUYMLSSM ONLLWOVdOld SAVM YOd WYSHLOS! 3.99 JO INSINZOV1dSIC
GNV NOLLOW TVOLLYSA GNV IWLNOZIMGH JO NOLLVLNSSSYdse WoOlHdVYS YZ auNdls

WHSHLOS! 45099 30 LNAWS0VIdSIG WOILYAA

Millia VM Le, =
fe it

3 2 ,
= —— SS —— EE fee ——— =e
= = SS
ES — —
=
a : _ —

FYNSSTYd NOILVGENLYSd © NOILOW ONIMOHS NOILOAS WOILUAA 1S3M-LSV3

ee
O

WOLLOG UVSN NOILOW TVINOZIYOH

Be eee

3OVSUNS UYVIN NOILOW WLNOZIYMOH

Hea Gee) hee

h> a2 + h? k> pas Ss (3h)

where 4 e a, (stability due to stratification)
i, ee
VI. SUMMARY AND CONCLUSIONS

An analysis for the spectral periods of internal waves was made from
temperature readings, comprising nearly 1,200 bathythermograms taken at
half-hourly intervals during Operation STANDSTILL. The analysis shows no
significant power associated with periods from 1 to 20 hours, but a con=
centration of the power in the band of periods between 20 and 26 hours. If
there is no aliasing from periods of less than 1 hour, it may be concluded
that the analysis of the temperature data fails to show that internal waves
of periods of less than 20 hours cerry any appreciable amount of power at the
location of the operation.

Two possible causes are suggested for the predominance of spectral periods
between 20 and 26 hours: formation of eddies on the southern edge of the
Gulf Stream due to instability gives periods of 21.7 hours, but the diameter
of the eddies is too small to account for the observed isothermal changes;
internal motion due to forced inertial oscillation is suggested, with a period
somewhat more than 21.7 hours. The latter oscillation apparently is in re-
sonance with the diurnal tidal components, and as a result the spectral power
is concentrated in the observed periods.

See Fjeldstad, J. E., Interne Wellen, Geofysiske Publikasjoner, Vol. X;
Now 6, Osilo, 1933.

19

BIBLIOGRAPHY

HAURWITZ, BERNHARD. Dynamic meteorologye New York: McGraw-Hill.
365 pe 191.

=» - « Internal tidal waves in the ocean. Technical Report submitted to
Geophysics Branch, ONR under contract N6 onr-27701 (NR-083-00)), Refer=
ence noe 53-69, Woods Hole Oceanographic Institution, Woods Hole, Massa=
chusetts. Unpublished. 3 p. 1953. ae

PIERSON, We Joy JRe A unified mathematical theory for, the analysis, propagation,

and refraction of storm generated ocean surface waves, parts I and IT,.

March 1, 1952 and July 1, 1952. New York, New York University. College

of Engineering. Department of Meteorology. Prepared for Beach Erosion

Board, Contract noe W h9-055-eng. 13 Office of Naval Research, Contract

no. Nonr-285 (03). 336 & 125 p. 1952.

Bye Wand MARKS, WILBUR. The power spectrum of ocean=wave records, Trans=
actions of the American Geophysical Union, vols 33, pe 834—Bh, 1952.

TUKEY, J. W. The sampling theory of power spectrum estimates, Symposium
on Applications of Autocorrelation Analysis to Physical Problems, Woods
Hole, Massachusetts, 13-1) dune 1919, po hf-67, 1950.

U. S. HYDROGRAPHIC OFFICE, Effects of weather upon the thermal structure
of the ocean, Hydrographic Office Miscellaneous 15360, 81 pe 1952.

VAN MIEGHEM, J. M. "Hydrodynamic instability", pe h3l-l53. In: Malone, TF.

ad

ede, Compendium of Meteorology. Boston: American Meteorological Societys

133) De 19516

VON ARX, We S., BUMPUS, D. Fo, and RICHARDSON, We Se. Short term fluctuations
in the structure and transport of the Gulf stream system. Technical Report
submitted to Geophysics Branch, ONR under contract N6 onr=27701 (NR=083-00})
and Nonr=769(00). Reference now 5-76, Woods Hole Oceanographic Institution,
Woods Hole, Massachusetts. Unpublished, 23 pe 1950 Nanhai.

WORTHINGTON, Le Ve The downstream increase in volume of the Florida current.
Technical Report submitted to Geophysics Branch, ONR under contract N6 onr-
27701 (NR-083-00)1). Refereace noe 55-3, Woods Hole Oceanogravhic Institution,
Woods Hole, Massachusetts. Unpublished. 9 p. 1955. Pere wa

9-1 °O°H *H
THALSGN¥IS
uoljosedQ woly sean,
Jousesuy yo sisAjouy

winajradg semod :ejtis *}

ABojasoasyow °S

AydosBounes9 *y
AydpsBounes9

* "A°N ‘upe8s0 s1uDjry °F
ssAjouy

winsyosdg 28M0 4 °F

S@ADA jOUIS{U] >}

9S-AL °O PH
TULSGNVLS
uojjpsedQ wo1y saad
jousesuy jo sysAjouy

uinajoadg seM04 -efst) °F

ABojoscajyey °S

4ydn sBoune20 *y
4ydnsBoune2Q

© °2°N ‘ub920 Jyunjiy °f
sjsAjpouy

winsgoedg 18M0 ¢ °F

SOADM [DUJO4U] *

*saney 9% pud QZ usemjeg
[PAsequy OYE Uy 4eMOd 50 UO!NUeDUOD eu 204
suROD3D 04 pejuesesd esp SUcSPa: OM *sSpoyseu
unajseds yemod Aq pezA;pun e19m TFULSGNYLS
uorypsedQ uo uerpt SuotDAasosgo ydo sBoueysAuiog
Ajanoy-jjoy jo sBuyppes einyosadwes poydejeg

Aydo sBoijqrg

“(97-44 °O °H)

rages | *seanBiy £ *dzt *SS6L 4eqo1DQ ‘ITNLS

“GNVLIS NOILVYSdO WONS SSAVM IVNYSL
“NI SO SISATIVNY WNNLOSdS- YAMOd

31430 214ydosBozpdApy AaDN °g °F)

°sanoy 9% pub (QZ weemjeq
PPAseguL @yy Uy 4dMOd 4O UOJ{D4ZUBILDD EY} 410}
guno33D 0} pesueseid ein Ssucspe! AKA) “"spoyssu
uinsyazeds senod Aq pezAjoun ss8M 11 SGNVLS
uO1jp1sdQ uC ue D4} sudsjDAJeSgO UdayGouseusAyog
Ajsnoy-sjoy je sBujppoes ein soduiey pesejes

Aydp i601) q}

(97-41 "OCH

reyqos 1 ‘seanBiy 2 “dz, “SS6L 2eq01>Q “TLS

-GNVLS NOILVYSdO WOYA SSAVM IVYNYSL
"NI 4O SISAIVNY WNY1D3dS- YAMOd

8213j0 apydosBopApy Aon "Ss °F)

9-YL °O °H MH
THULSGNVIS
uojppsedQ) wioay SHAD
Jouseyu) yo sysAjpuy

winagoedg 4amodg +efti) *|

ABojoscayey °S

AydpsBounesQ *y
AudposBoupe2Q

2 *S°N “upe20 JI sud} IW °E
sishjouy

uinsyoedg 19M0 4 °Z

SOADM jDUAE4UL *|

9o-H1L °O °H NH
THLSGNV LS
Uo}j;DsedQ wo1 sandy
jousozuy yo sysAjpouy

winsgaedg 4emod :ej4sty *}

ABojoscaey °S

Aydp sBouve29 *y
AydosBounesg

2 °S°N ‘ube20 Jub] ry °E
sisAjouy

winsyvedg somo q °Z

SOADM JOUsO4UI *|

*sanoy 9Z pup QZ ueeMmseq
JPAsoyuy OY4s Uy 4eMOd 4O UOIJDIZUeDQUOD OYs 404
junossp O4 peyueseid 81D SUOSDO1 OM] “Spouse
tunsyoeds aemod Aq pezAjoun 18M ATWISONVIS
uolpsedQ uo ua pg Sud! Asesgo YydosBowseysAuyDg
Ajinoy-yjoy yo sBulpoes esnyossduisy poy39]9¢

Aydo sBorqig

“(92-41 *O °H)

rayqos | ‘seanBiy Z “*dz] “SS6L 42q0120 “TTILS

“GNVLS NOILVYAdO WOYS SAAVM WNUSL
“NI dO SISATIVWNY WNYeLldadS- ¥AMOd

©219j0 2{ydosBoupAyy AADW °g °f)

*sanoy 9% pup QZ useMmjeq
jPAseyur ey; Uy 4eMOd jo UO!JOAjUe>UCD ey 404
fUNDIID Of pesuesesd ei SuoSsDes OAL “Spoupeut
wnajaeds semod Aq pezdjoun e18M VTLSONVIS
ua1jD4edQ UO User Ds SUO;DAIESgo YdosBowseYysAYDG
Aysnoy-j joy yo sGujpoes eunjpsedwes peysejeg

Aydo sBoyiqig

(90-41 °O *H)

rayqoa | ‘seunByy £ “*dz_ “"SS6L 42qG019Q “TTILS

“GNVLS NOILVYSdO WONS SSAVM WNAYAL
"NI SO SISAIVNY WNYLdadS- Y3sMOd

931339 214dDsBospAyy AADWY °S °f)

9-HL “O °H “HH
TIULSGNYLS
uoypsedg wo seany
JPuseju] yo sisAjpuy

wnajoadg semog :ejtis *}

ABojasoayop °S

AydosBounes9 *F
AydosBoune3Q

> A°N ‘upe5Q ShuDjry °E
sisAjpuy

winsyosdg 28M0 4 °F

SSADM JDUIS{U] >}

FAL °O MMH
THLSGNVLS
uoyiDsedQ wos Saad,
JPusesul yo sysAjDuy

uinsyasdg samod 72441) °F

ABojosc83;0yW °¢

4ydpsBounssQ *y
AydosBoune2Q

= °2°N ‘uD92Q aHUD}Iy “E
sisAjpuy

uinayoedg seM0 ¢ °%

SCADA [DULO3UI

*saney QZ pud QZ usemjeq
[PAsezuy EUs UY 4eMOd jo UOTD4YUeDHOD eLY 404
$uN0D9D 04 pejueseid e1D SucsPe: OM! *Spoujeu
uinsgaeds semod Aq pezAioun e19m FFU LSGNYLS
uoryp4edQ uo UexDL SUOLIDALESGO uso sSoueusAuing
Ajsnoy-jjou jo sBurppes ainpsadues peyoejes

Aydp sBoijqrg

“(9T-YL °O °H)

reyqos | *seanByy Z “dz, “SS6L 4840520 ‘ITNLS

“GNVLS NOILVYAdO WOY¥S SSAVM WWNYSL
"NI SO SISATYNVY WNYLDAdS- YAMOd

21330 214ydosBozpApy Kady) °g °F

°sanoy 9% pur (QZ usemjeq
JPAseyuL @Yyy Uy 4oMOd jo UO}{D4jUSDIOD 943 104
§3UNOS3ID OB pesueseid @ip SuoSsDes QA) *spoysow
winsyaeds semod Aq pszhjoup sJeM TI LSGNVLS
UO}jD18dO UO USD} SUd}yDAIeSgO YdnyBousaysAyIDG
Ajsnoy-ypoy jo sBuyppes eunypsadwas pesz2e}e¢

Aydo i601) q)
“(9T-H1 °O °H
rayqos 4 ‘seanBiy 2 “dz “SS6L 2990120 ‘TTILS
-GNVLS NOILVYSd0 WOUs SASAVM TWNYSL
"NI SO SISATIVNY WnNdldadS- YaMOd
821530 214dDsBoupApy AaDN °6 °F)

9o-HL °O °H “IH
THLSGNVLIS
uojposedy wor SAD
Jouseyuy yo sjsAjpuy

wunajredg 4emMOq 2@f41} °}

ABojouceyey °S

Aydo sBoun63Q *p
A4udoiBouve3G

© *S°N ‘UDe20 JIuD]iy °E
sishjpuy

uinsg2edg 19M0 ¢ °T

SEADM [DUO 4UY >}

9-H °O °H MH
THULSGNVLS
uoysp4edQ woiy Sead
jouseyuy yo ssAjpuy

WINIGIBGS JEMO 4 iOjsly °Y

ABojos0s eZ °S

Aydo sBounesQ *y
AydosBounesQ

© *2S°N ‘up850 JIjubj ry °F
sisAjpuy

unsyoedg semo dq °Z

SOADM jDUIE4UY °|

*sunoy 9Z puod QZ uceemseq
JPAsoyuy O44 Uy 4emMod yo UOlWDsyUeDU0D E44 405
junorzSD Oj pejueseid 81D SUOSDO1 OM "*SpPOyjous
uinsyseds samod Aq pezdAjpun 218M VTISONV LS
uoippsJedQuo usrpy SUO}4D Alas gO ydo sBowseyyAuyog
Ajanoy-3joy jo s6ujpoes einyossduis, pe4y3e)e¢

Aydo sB01)/q1q

(9Z-UL *O °H)

seyqos | ‘sesnBiy Z “dz, “SSL 4990190 ‘TTILS

“GN¥LS NOILVYSdO0 WOYS SAAVM WWNYAL
“NI SO SISATIVNY WNdldadS- YaM0d

821330 21ydosBosphyy AAD °g °f}

‘sanoy 9% pup QZ ueemjoq
jPAsequi Oy; Uy 4eMOd JO UO!JOJUSDUOD eYs 104
fuNEeDID a4 pesuesesd @1D SUOSDOs OA] “Spouseu!
uinajgeds semod Aq pezdAjpun elem FTWISGNVIS
u01jDsedQ UO UarxDs SUOI{OAJOSgoO YdosBouwseYsAyDG
Ajsnoysyjoy yo sBujpoes esnjosedwsy pes29j0¢

Aydo abo qig

“(9%-aL °O *H)

rerqoa | ‘seunByy £ “dz, “SS6L 20qG0120 “TTILS

“GNVLS NOILVuSdO WOYS SHAVM “IWNYAL
“NI SO SISATIWNY WNYLDSdS- YaMOd

831}j9 214dDsBoupAyy AaDWY °S °f)